Lean six sigma (green belt)new

LEAN
SIX SIGMA
NATIONAL PRODUCTIVITY ORGANIZATION (NPO), LAHORE
MINISTRY OF INDUSTRIES, PAKISTAN
1
SL S

MODULE # 1
2 LEAN SIX SIGMA: AN
OVERVIEW
 Understanding about Lean & Six Sigma
 Five Lean Principles
 Types of Waste
 Eight Sources of Waste
 A Simple Lean Tool: “5S”

EVOLUTION OF QUALITY
FIELD
Product Insp. to Process to System to Culture to Performance
Control (Opr Mgt ) Change
TQM+ – Wave II
Lean Six
Sigma
Six Sigma
Knowledge
Mgt.
IT
TQM – Wave I
HRM
GROUP
DYNAMICS
Teams
Efficiency
BPR
TPM
JIT/MRP
QA
ISO9000
OPR MGT.
QC
SPC
Quality
Circles
Inspection/
Testing
Metrology
3

WHAT IS LEAN?
4
“A systematic approach to identify and eliminate waste (and non
value-added activities) through continuous improvement by flowing
the product at the pull of the customer in pursuit of perfection.”
Lean Thinking is all about continuous waste elimination !
LEAN is Delivering value to Customers with shortest turn around time

SIX SIGMA DEFINITIONS
 A Management driven, scientific methodology for
product and process improvement which creates
breakthroughs in financial performance and
Customer satisfaction.
Source: Motorola
 A methodology that provides businesses with the
tools to improve the capability of their business
processes. This increase in performance and
decrease in process variation lead to defect reduction
and improvement in profits, employee morale and
quality of product.
Source: ASQ
5

COMPARISON OF
LEAN & SIX SIGMA6
Six Sigma was developed by Motorola in the 1980s to systematically
improve quality by elimination of defects.
SIX SIGMA LEAN
Objective Deliver value to customer Deliver value to customer
Theory Reduce variation Remove waste
Focus Problem focused Flow focused
Assumptions  A problem exists
 Figures and numbers are
valued
 System output improves if
variation in all processes
inputs is reduced
 Waste removal will
improve business
performance
 Many small improvements
are better than system
analysis
Six Sigma is a data driven philosophy and process resulting in dramatic
improvement in products/service quality and customer satisfaction.

5 PRINCIPLES OF LEAN
7 In “Lean Thinking” – as summarized by James Womack & D. Jones
1 SPECIFYING VALUE
“Value is only meaningful when expressed in terms of a
specific product or service which meets the customer
needs at a specific price at a specific time.”
2 Identify and Create
Value Streams
“A Value stream is all the actions currently required to
bring a product from raw materials into the arms of the
customer.”
3 Making Value Flow
“Products should flow through a lean organization at the
rate that the customer needs them, without being
caught up in inventory or delayed.”
4
“Only make as required. Pull the value according to the
customer’s demand.
Pull production not
Push
5 Striving for
perfection
“Perfection does not just mean quality. It means
producing exactly what the customer wants, exactly
when the customer requires it, at a fair price and with
minimum waste.

LEAN MANUFACTURING:
ELIMINATING THE WASTE8
Waste caused by overstressing people,
equipment, or systems
Waste due to unevenness or variation
 Type–I Muda: Non-Value added, but necessary
for the system to function
 Type–II Muda: Non – Value added and
unnecessary for the system to function; the first
targets for elimination
(Unreasonableness)
Muri
Mura
(Inconsistency)
Muda (Waste)

LEAN MANUFACTURING:
ELIMINATING THE WASTE9
Types of Waste (Muda)
 Transportation
 Inventory
 Motion
 Waiting
 Overproduction
 Over processing
 Defect
 Non-utilized people
Transportation

TYPES OF WASTE (MUDA)
10
Waste (Muda)
Total lead time through the value chain
1. Are they equal or not?
2. If not; Which is the most significant source of
waste?

Which is the most significant source of waste?
Producing
TOO much
ADDITIONAL
transportation cost
Producing
TOO much
to sort, handle and store.
Overproduction is the disease, Defects are the cause?
TYPES OF WASTE (MUDA)
11

5S – A Simple “LEAN TOOL”
12
Implementing the 5S is often the first step in
Lean Transformation
5S – A Framework to create and maintain your workplace
1. S: SORT (Organization)
Distinguish between what is and is not needed
2. S: SET IN ORDER (Orderliness)
A place for everything and everything in its place
3. S: SHINE (Cleanliness)
Cleaning and looking for ways to keep it clean
4. S: STANDARDIZE (Adherence)
Clearly define Tasks and Procedures
5. S: SUSTAIN/SYSTEMIZE (Self-Discipline)
Stick to the rules, conscientiously

5S GAME
13
The “Numbers Game” is an
exercise that illustrates the
power of 5s.
The game consist of 4 quick
rounds. You must not look at
the sheets until instructed and
must finish when time is up

5S GAME: ROUND#1
14
 We will apply 5S to a workplace and measure
the improvement in executing our job.
 During each 30 second shift, your job is to
strike out the numbers 1 to 49 in order
 The first page of numbers represents our
current workplace
 Ready… Set…

(Cont…)15
Decide what is needed and what is not,
and dispose of all items that are not necessary
1 SORT / SIFTING
CAR
PARKING
AREA
Should these barrels
be in car parking area?

5S GAME: ROUND#2
16
 Japanese concept for
house keeping
 Sort (Seiri)
 Straighten (Seiton)
 Shine (Seiso)
 Standardize (Seiketsu)
 Sustain (Shitsuke)
 The first “S” is Sort
 We have removed
numbers between 50
and 90 which are not
needed
 Ready… Set…
 What sort of
improvement does this
yield?

(Cont…)17
2 STRAIGHTEN / SET IN
ORDER
Provide orderly storage in the right place for
all necessary items so that they can be easily
found and used when needed.
EQUIPMENT
STORAGE AREA
Easy to determine
equipment location

5S GAME: ROUND#3
18
house keeping
 Sort (Seiri)
 Straighten
(Seiton)
 Shine (Seiso)
 Standardize (Seiketsu)
 The second “S” is
Straighten or Set in Order
 We have installed a rack
system to help locate
the numbers.
 Numbers go from
bottom to top, left to
right
 Ready… Set…
 What sort of improvement
does this yield?

(Cont…)19
3 SHINE / SWEEPING
Maintain a clean worksite at all times in order
to make work easier, safer, healthier and more
satisfying
I am motivated to
work in this standards
Service
Workshop

(Cont…)20
4 STANDARDIZE
Continuously keep work
area orderly and clean

5S GAME: ROUND#4
21
house keeping
 Sort (Seiri)
 Straighten (Seiton)
 Shine (Seiso)
 Standardize
(Seiketsu)
 The fourth “S” is
Standardize
 We’ve created a system of
ordering the numbers from
lowest to highest from left
to right and top to bottom
 We’ve put one number in
each box to standardize
 Ready… Set…
 What sort of improvement
does this yield?

(Cont…)22
5 SUSTAIN / SELF-DISCIPLINE
Make it habit to engaging 5S activities
daily basis by establishing standards.
 The fifth “S” is Sustain
 This is your challenge: Sustain your lean activities
 Often the hardest to achieve

MODULE # 2
23
SIX SIGMA: AN OVERVIEW
 What is Six Sigma?
 Different opinions on the definition of six sigma
 Six Sigma is a Philosophy
 Six Sigma is a Set of Tools
 Six Sigma is a Methodology
 Six Sigma as a Measure
 Six Sigma as a Metric
 Six Sigma Structure

WHAT IS SIX SIGMA?
 In a narrow sense…
 A metric based on Statistical Measure called
Standard Deviation
 In a broader, business sense…
 WORLD CLASS QUALITY providing a BETTER
product or service, FASTER, and at a LOWER
COST than our competitors.
 VARIATION… “the enemy of the customer
satisfaction”
24

WHAT IS SIX SIGMA?
 DIFFERENT OPINIONS ON THE DEFINITION OF SIX SIGMA:
 Six Sigma is a PHILOSOPHY:
 This is generally expressed as y = f(x).
 Six Sigma is a SET OF TOOLS:
 The Six Sigma expert uses qualitative and quantitative
techniques to drive process improvement.
 Six Sigma is a METHODOLOGY:
 DMAIC Vs DMADV
 Six Sigma is a METRIC:
 it uses the measure of sigma, DPMO (Defect Per Million
Opportunities), RTY (Rolled Throughput Yield) etc.
 Six Sigma is a MEASURE:
 Short Term Vs Long Term
25

SIX SIGMA FRAMEWORKS
26
SIX
SIGMA
Lean Six
Sigma DFSS
VARIATION
Defects
Cost of Poor
Quality
WASTE / SPEED
Cycle Time,
Delivery
Cost of Operation
RELIABILITY &
ROBUSTNESS
Design Features
DMAIC DMAIC DMADV
SIPOC,
CTQ, SPC,
FMEA, DOE,
QFD, CoQ,
ANOVA,
Hypothesis,
Regression,
MSA (R & R)
5S, Value
Mapping,
Time Study,
TPM,
Cellular
Prod.,
Takt Time,
Poke Yoke
VOC, QFD,
FMEA, CTQ,
Gage R & R,
DOE,
Reliability
Analysis, SPC,
Systems
Engineering
PROGRAM
FOCUS /
THEME
METHODOLOGY
TOOLS

DMAIC METHODOLOGY
27
 DEFINE: "What is important to the business?"
 The problem is defined, including who the customers are and what they want, to
determine what needs to improve.
 Expected benefits for the project sponsor & Time line
 MEASURE: "How are we doing with the current process?"
 The process is measured, data are collected, and compared to the desired state.
 ANALYZE: "What is wrong with the current process?"
 The data are analyzed in order to determine the cause of the problem.
 IMPROVE: "What needs to be done to improve the process?"
 The team brainstorms to develop solutions to problems; changes are made to the
process, and the results are measured to see if the problems have been eliminated.
If not, more changes may be necessary.
 CONTROL: "How do we guarantee performance so that the improvements are
sustained over time?"
 If the process is operating at the desired level of performance, it is monitored to
make sure the improvement is sustained and no unexpected and undesirable
changes occur.

SIX SIGMA IS A METRIC
WHAT IS SIGMA LEVEL?
A metric that indicate how well a process is performing. A
higher sigma level means higher performance . A Statistical
measure of the capability of a process.
28
1. Defects
2. Defects Per Unit (DPU)
3. Parts Per Million (PPM)
4. Defects Per Million Opportunities (DPMO)
5. Yield
6. First Time Yield
7. Rolled Throughput Yield (RTY)
8. Sigma Level
Each of these metrics serves a different purpose and may be used at different
levels in the organization to express the performance of a process in meeting the
organization’s (or customer’s) requirements.

SIX SIGMA: METRICS …
29
DPU
(Defects / Unit)
(# of Defects / # of Units)
Say:
10 Defects, 100 Pairs
DPU = 10/100 = 0.1 (10%)
DPO
(Defects / Opportunity)
(# of Defects) / (# of Units
X # of Defect Opportunities
/ Unit)
Say:
10 Defects, 100 Pairs,
2 Opportunities / Carton
DPO = 10/(100 X 2) = 0.05
or 5% for each type

SIX SIGMA: METRICS …
30
DPMO
(Defects / M.
Opportunities)
DPO X 106
Say:
10 Defects, 100 Pairs
2 types of defects
DPMO = 0.05 X 106 = 50,000
SIGMA
Consult Z–Table or Excel
Sigma Level
Yield =1–DPO =1–0.05 = 95 %
From M.S. Excel:
=Normsinv(%Yield)+1.5
50,000 DPMO = 3.145σ

SIX SIGMA IS A MEASURE
1.5 Sigma
Shift
Theory
31

SIX SIGMA STRUCTURE
32
Quality Council / Steering Committee
Champions
Master
Black Belt
Black Belt Black Belt
Green Belt Green Belt Green Belt Green Belt
HOD’S /
Owners
Sponsors
Process Owner
Coach
Trainers
Team
Leaders
Team
Members
Project
Managers

MODULE # 3
33 DMAIC METHODOLOGY:
DEFINE PHASE
 Project Charter
 IPO & SIPOC Diagram
 Process Flow Diagram
 Lean Process Metrics
 Cost of Quality

PROJECT CHARTER
34 Project Title Project Title
Business Case  Why should you do this project?
 What are the benefits of doing this project?
Problem Statement What is the problem, issue and/or concern?
Goal What are your improvement objectives and targets?
Metrics (CTQ’s)  PRIMARY Metric(s): Key measures to be used for the
objectives
 SECONDARY Metric(s): Those measures which indicates
impacts on secondary concerns and which indicates that
problem is not shifted to other key areas.
Project Scope  What authority do you have?
 Which processes/products you are addressing?
 What is not within this project?
Project Team  Who are the team leader, sponsor, and members?
 What are their roles and responsibilities in this project?
Project Plan How and when are you going to get this project done (DMAIC
stages)
Communication
Plan
 What are your interfaces with each other?
 What are your meeting & reporting times?

WHAT IS A PROCESS?
(IPO DIAGRAM)
 An Input-Process-Output (IPO) diagram, also known as a general process
diagram,
 provides a visual representation of a process by defining a process and
demonstrating the relationships between input and output elements.
 The input and output variables are known as ‘factors’ (X) and ‘responses’ (Y)
, respectively.
36
INPUTS OUTPUTS
BILLING
PROCESS
Data Entry Method
Amount of Personnel
Training
Method for obtaining bill
from information
Time to complete a bill
Number of errors / bill

TYPES OF PROCESS MAPS:
SIPOC DIAGRAM
Suppliers Inputs Process Outputs Customers
6) Who are the
Suppliers of the
Inputs?
5) What
are the
Inputs of
the
Process?
2)a. What is the
start of the
process?
1) What is the
process?
2) b. What is the
end of the
process?
3) What are
the outputs
of the
process?
4) Who are
the customers
of the
outputs?
37

TYPES OF PROCESS MAPS: SIPOC DIAGRAM…
 Example SIPOC Diagram of Husband making wife a
cup of tea.
38

PROCESS FLOW DIAGRAM
(FLOW CHART)
 A Flowchart is a diagram that uses graphic symbols to represent the
nature and flow of the steps in a process / system.
 Deciding when & where to collect data
FEW SYMBOLS USED IN FLOW DIAGRAM
Process Symbol
“An Operation or Action step”
Terminator Symbol
“Start or Stop Point in a process”
Inventory / Buffer
“Raw Material / Finished Goods Storage”
Inventory / Buffer
“Partial Finished Goods
“Work In Process” Storage” Document Symbol
“A Document or Report”
Database Symbol
“Electronically Stored
Information”
Flow Line
Decision Point
39

PROCESS FLOW DIAGRAM ….
1.What you THINK it is …
2.What it ACTUALLY is…
3. What it SHOULD be…
40

PROCESS MAPPING LEVELS
41  LEVEL–1: The Macro Process Map, sometimes called a Management
Level or viewpoint.
 LEVEL–2: The Process Map, sometimes called the worker level or
viewpoint. This example is from the perspective of the pizza chef.
 LEVEL–3: The Micro Process Map, sometimes called the Improvement
level or viewpoint. Similar to a level–2, it will show more steps and
tasks and on it will be various performance data; yields, cycle time,
value and non-value added time, defects, etc.

TYPES OF PROCESS MAPS
42
THE LINEAR FLOW PROCESS MAP
THE DEPLOYMENT FLOW or SWIM LANE PROCESS MAP

LEAN PROCESS: Value Added
& Non–Value Added43
Value Added Activity
 Transforms or shapes material or information or people
 And it’s done right the first time
 And the customer wants it
Non-Value Added Activity – Necessary Waste
 No value is created, but cannot be eliminated based on
current technology, policy, or thinking
 Examples: project coordination, company mandate, law
Non-Value Added Activity - Pure Waste
 Consumes resources, but creates no value in the eyes of
the customer
 Examples: idle/wait time, rework, excess checkoffs

LEAN PROCESS METRICS
 Processing times or activity time: how long does the worker or
process spend on the task?
 Capacity: how many units / customers can the worker or process
make / deal per unit of time
 Bottleneck is the process step with the lowest capacity
 Process Capacity is the capacity of the bottleneck
 Cycle time is the time interval between the completion of two
consecutive units (or batches)
 Flow rate (Throughput rate) is the output rate that the process is
expected to produce
 Flow Time (Throughput time) = The amount of time for a unit /
customer to move through the system
 Inventory: The number of flow units / customers in the system
 Utilization is the ratio of the time that a resource is actually being
used relative to the time that is available for use
 Work Load/Implied Utilization = Capacity requested by demand /
Available Capacity
 Takt Time: (Available Time) / Demand
44

QUALITY COST
 Feigenbaum defined quality costs as:
“Those costs associated with the definition, creation, and
control of quality as well as the evaluation and feedback of
conformance with quality, reliability, and safety
requirements, and those costs associated with the
consequences of failure to meet the requirements both
within the factory and in the hands of customers.”
“QUALITY IS FREE” (Crosby)
45

COST OF QUALITY
 COST OF ACHIEVING GOOD QUALITY
 PREVENTION COSTS
 The cost of any action taken to investigate, prevent or reduce the risk of a non-
conformity
 Include quality planning costs, designing products with quality characteristics,
Training Costs, etc.
 APPRAISAL COSTS
 The costs associated with measuring, checking, or evaluating products or services
to assure conformance to quality requirements
 Include inspection & Testing Costs, Test Equipment Costs, Operator Costs, etc.
 COST OF POOR QUALITY
 INTERNAL FAILURE COSTS
 The costs arising within the organization due to non-conformities or defects
 include scrap, rework, process failure, downtime, and price reductions
 EXTERNAL FAILURE COSTS
 The costs arising after delivery of product or service to the customer due to non-
conformities or defects
 include complaints, returns, warranty claims, liability, and lost sales
46

MEASURING AND REPORTING
QUALITY COSTS
INDEX NUMBERS
ratios that measure quality costs against a base
value
LABOR INDEX
ratio of quality cost to labor hours
COST INDEX
ratio of quality cost to manufacturing cost
SALES INDEX
ratio of quality cost to sales
PRODUCTION INDEX
ratio of quality cost to units of final product
47

MODULE # 4
MEASURE PHASE
 Types of Data
 Data Collection (Check Sheet)
 Pareto Analysis
 Cause & Effect Diagram & Matrix
 Descriptive Analysis
 Qualitative Data Analysis
 Quantitative Data Analysis
 Process Capability Studies (Cp,Cpk)

TYPES OF DATA
49  Attribute Data (Qualitative)
 Is always binary, there are only two possible values (0, 1)
1. Yes, No
2. Success, Failure
3. Go, No Go
4. Pass, Fall
 Variable Data (Quantitative)
Discrete (Count) Data:
 Can be categorized in a classification and is based on
counts.
1. Number of defects
2. Number of defective units
3. Number of Customer Returns
Continuous Data:
 Can be measured on a scale, it has decimal subdivisions
that are meaningful
1. Time, Pressure
2. Money
3. Material feed rate

CHECK SHEET
 A check sheet is a Form, in Diagram or Table format, prepared in
advance for Recording/Collecting Data. You can thus gather necessary
Data by just making a Check mark on the Sheet.
50
TYPES OF CHECK SHEET
DEFECTIVE ITEM (ATTRIBUTE)
CHECK SHEET:
VARIABLE CHECK SHEET:
DEFECTIVE LOCATION CHECK
SHEET:

PARETO ANALYSIS
 A bar graph used to arrange information in such a way that
priorities for process improvement can be established.
Count
Percent
Fault Desc.
Count 43 107
Percent 31.9 30.4 9.3 7.4 4.4 4.3 3.5
707
2.0 1.9 4.8
Cum % 31.9 62.3 71.6 79.0 83.3 87.7
674
91.2 93.2 95.2 100.0
205 164 97 96 78 45
Other
M
isc. Defects
Trim
Not As
Specified
Raw
Edge
Seam
Const. Not As
Specified
O
pen
Seam
Uneven
St.
Puckering
Skip
St.
Broken
St.
2500
2000
1500
1000
500
0
100
80
60
40
20
0
Pareto Chart of Fault Desc.
 The 80–20 theory was first developed in 1906, by Italian economist,
Vilferdo Pareto, who observed an unequal distribution of wealth and
power in a relatively small proportion of the total population.
 Joseph M. Juran is credited with adapting Pareto’s economic
observations to business applications.
 Separates the "vital few" from the "trivial many" (Pareto Principle)
51
80 – 20 Rule:
 80% of your phone calls go to
20% of the names of your list
 20% of the roads handle 80% of
the traffic
 80% of the meals in a restaurant
come from 20% of the menu
 20% of the people causes 80% of
the problems

PARETO ANALYSIS …
 Pareto Chart Using Minitab– EXAMPLE:
Suppose you work for a company that
manufactures motorcycles. You hope to reduce
quality costs arising from defective
speedometers. During inspection, a certain
number of speedometers are rejected, and the
types of defects recorded.
You enter the name of the defect into a
worksheet column called Defects, and the
corresponding counts into a column called
Counts.
You know that you can save the most money by
focusing on the defects responsible for most of
the rejections.
A Pareto chart will help you identify which
defects are causing most of your problems.
Open the worksheet EXH_QC.MTW
52

CAUSE & EFFECT DIAGRAM
 Step 1: Identify the problem and enter in effect box
 Step 2: Draw in the spine of fishbone
 Step 3: Identify Main Causes
 Step 4: Identify sub-causes influencing the effect
 Step 5: Identify detailed causes and analyze diagram
53
OPEN THE FILE: SURFACEFLAWS.MTW
The “Y”
The
“Problem”
The “X’s”
“Causes”

DEFINITION OF X–Y MATRIX
 The (X–Y) CAUSE & EFFECT Matrix is:
 A tool used to identify potential X’s and assess their relative
impact on multiple Y’s (including all Y’s that are customer
focused)
 Based on the team’s collective “opinions”
 Created for every project
 Updated whenever a parameter is changed
 To summarize, the X–Y is a team based prioritization tool for the
potential X’s
 WARNING! This is not real data, this is organized brainstorming!!
At the conclusion of the project you may realize that the things
you thought were critical are in fact not as important as was
believed.
The X–Y Matrix is this Prioritization Tool!
54

THE X–Y MATRIX …EXAMPLE: Let’s take the example of the newspaper printing process. After
transforming the VOC we find the CTQ (Y’s) as below:
 Clearly readable print
 Good quality photo
 Harmless to health
Upon brainstorming, the input process parameters (X’s) have been found as below:
 Good quality ink
 Less vibration during operation of printing press
 Paper quality
For the above sets of X’s and Y’s the X-Y matrix table will look like the example
below:
 Wherever there is a strong relation between X’s and Y’s, put 9. For weak relations put 3 or 1. Keep
the intersection field blank if there is significant relation.
 Weighted sum for “Good quality ink” is calculated as 15*9 + 10*9 + 10*1 = 235.
55
Output Parameters (Y’s)
Clearly readable
print
Good Quality
Photo
Harmless to
health
Weighted
Sum of X’s
Weightage of Y’s 15 10 10
Input
Parameters
(X’s)
Good quality ink 9 9 1 235
Vibration less
operation of printing
press
9 3 165
Paper quality 1 1 9 115

SIX SIGMA
METHODOLOGY
DMAIC
MEASURE PHASE
SIX–SIGMA STATISTICS
56

WHAT IS STATISTICS?
1. Collecting Data
 e.g. Survey
2. Presenting Data
 e.g., Charts & Tables
3. Characterizing Data
 e.g., Average
57
Why?Data
Analysis
Decision-
Making
© 1984-1994 T/Maker Co.

TYPES OF DATA58
 Attribute Data (Qualitative)
 Is always binary, there are only two possible values (0, 1)
1. Yes, No
2. Success, Failure
3. Go, No Go
4. Pass, Fall
 Variable Data (Quantitative)
Discrete (Count) Data:
 Can be categorized in a classification and is based on counts.
1. Number of defects
2. Number of defective units
3. Number of Customer Returns
Continuous Data:
 Can be measured on a scale, it has decimal subdivisions that are
meaningful
1. Time, Pressure
2. Money
3. Material feed rate

DEFINITIONS OF SCALED DATA
59
TYPE OF
DATA
OPERATOR DESCRIPTION EXAMPLES
Nominal =, ≠ Categories Types of defects,
Types of colors
Ordinal <, > Rankings Severity of
defects: critical,
major, minor
Interval +, - Differences but
no absolute zero
Temperature of a
ship
Ratio / Absolute zero Pressure, Speed

STATISTICAL METHODS
60
STATISTICAL
METHODS
DESCRIPTIVE
STATISTICS
INFERENTIAL
STATISTICS
1. Involves
 Collecting Data
 Presenting Data
 Characterizing Data
2. Purpose
 Describe Data
1. Involves
 Estimation
 Hypothesis Testing
2. Purpose
 Make Decisions About
Population Characteristics

DESCRIPTIVE ANALYSIS OF
QUALITATIVE DATA61
QUALITATIVE DATA
TABLES GRAPHS NUMBERS
One Way Table
Two–Ways Table
.
.
.
N – Ways Table
Bar Chart
Pie Chart
Multiple Bar Chart
Component Bar
Chart
Percentages

DESCRIPTIVE ANALYSIS OF
QUANTITATIVE DATA62
QUANTITATIVE DATA
TABLES GRAPHS NUMBERS
Frequency Distribution
Stem and Leaf Plot
Histogram
Box and Whisker’s Plot
Center
Important
Points
Variation
Mean
Median
Mode
Trimmed Mean
Median
Quartiles
Deciles
Percentiles
Range
Inter-Quartile Range
Variance
Standard Deviation
Skewness
Kurtosis
Distribution
Open “Pulse.mtw” ; Conduct Descriptive Analysis on the pulse1 data.

BOX & WHISKER’S PLOT
USING MINITAB
CONSTRUCTING BOX PLOT (One Y):
You want to examine the overall durability of your carpet products.
Samples of the carpet products are placed in four homes and you
measure durability after 60 days. Create a Box Plot to examine the
distribution of durability scores.
Open worksheet Carpet.mtw
63
Constructing Box Plot: (One Y–with Groups)
You want to assess the durability of four experimental carpet products.
Samples of the carpet products are placed in four homes and you
measure durability after 60 days. Create a box plot with median labels
and color-coded boxes to examine the distribution of durability for each
carpet product.
Open the worksheet CARPET.MTW

NORMAL DISTRIBUTION
64
 Characteristics of the normal distribution:
 Continuous distribution - Line does not break
 Symmetrical distribution - Each half is a mirror of the other half
 Asymptotic to the horizontal axis - it does not touch the x axis and goes
on forever
 Unimodal - means the values mound up in only one portion of the graph
 Area under the curve = 1; total of all probabilities = 1
 Normal distribution is characterized by the mean and the Std Dev
 Values of μ and σ produce a normal distribution
...2.71828
...3.14159=
Xofdeviationstandard
Xofmean
:
2
1
)(
2
2
1








 


e
Where
x
xf e






 X

NORMALITY TEST
65
Open the worksheet CRANKSH.MTW
 NORMALLY TEST:
o Generate a normal probability plot and performs a hypothesis test
to examine whether or not the observations follow a normal
distribution. For the normality test, the hypothesis are,
o Ho: Data follow a normal distribution Vs H1: Data do not follow a
normal distribution
o If ‘P’ value is > alpha; Accept Null Hypothesis (Ho)

PROCESS CAPABILITY66
 The inherent ability of a process to meet the expectations of the
customer without any additional efforts. (or)
 The ability of a process to meet product design/technical
specifications
– Design specifications for products (Tolerances)
 upper and lower specification limits (USL, LSL)
– Process variability in production process
 natural variation in process (3 from the mean)
 Provides insight as to whether the process has a :
 Centering Issue (relative to specification limits)
 Variation Issue
 A combination of Centering and Variation
 Allows for a baseline metric for improvement.

TWO KINDS OF VARIABILITY
67
 Common Cause / Inherent variability:-
 Inherent in machine/process (design, construction and nature of
operation).
 Special Cause / Assignable variability: -
 Variability where causes can be identified.
 Assignable variability eliminated / minimized by Process Capability
Study.
FOR A CAPABLE PROCESS:
INHERENT + ASSIGNABLE < TOLERANCE
PROCESS CAPABILITY ANALYSIS…
 PROCESS CAPABILITY STUDY ASSUMPTIONS:
1. The performance measure data reflects statistical control when
plotted over a control chart (i.e.: X–Bar & Range Chart)
2. The performance measure data distributed normally.

Process Capability Index:
Cp -- Measure of Potential Capability
6variationprocess
variationprocess LSLUSL
actual
allowable
Cp


68
Cp = 1
Cp < 1
Cp > 1
LSL USL Cp measures the
relationship between the
tolerance width and the
total range of process
variation.
 Cp does not consider the
location of the mean and
therefore represents the
potential of the process to
produce characteristics
within specification.

Process Capability Index:
Cpk -- Measure of Actual Capability
69
“σ” is the standard deviation of the production process
Cpk considers both process variation () and process
location (X)
Cpk takes into account any difference between the design target and the
actual process mean.

PROCESS CAPABILITY ANALYSIS:
EXAMPLE (Minitab)70
The length of a camshaft for an automobile
engine is specified at 600 + 2 mm. To avoid
scrap / rework, the control of the length of the
camshaft is critical.
The camshaft is provided by an external supplier.
Access the process capability for this supplier.
Filename: “camshaft.mtw”
Stat > quality tools > capability analysis
(normal)
Process Capability Indices & Sigma Quality Level

EXERCISE (Minitab)–BOTH SUPPLIERS71
Histogram of camshaft length suggests mixed
populations.
Further investigation revealed that there are two
suppliers for the camshaft. Data was collected
over camshafts from both sources.
Are the two suppliers similar in performance?
If not, What are your recommendations?
FILENAME: “camshafts.mtw”

PROCESS CAPABILITY FOR
NON–NORMAL DATA72
 To address non – normal data is to identify exact type of
distribution other than normal distribution
 INDIVIDUAL IDENTIFICATION OF DISTRIBUTION
 Use to evaluate the optimal distribution for your data
based on the probability plots and goodness-of-fit
tests prior to conducting a capability analysis study.
Choose from 14 distributions.
 You can also use distribution identification to
transform your data to follow a normal distribution
using a Box–Cox transformation or a Johnson
transformation.

NON–NORMAL DATA73
EXAMPLE:
Suppose you work for a company that manufactures floor tiles and are concerned
about warping in the tiles. To ensure production quality, you measure warping in
10 tiles each working day for 10 days.
The distribution of the data is unknown. Individual Distribution Identification allows
you to fit these data with 14 parametric distributions and 2 transformations.
Open Worksheet: Tiles.mtw

NON–NORMAL DATA74
Distribution ID Plot for Warping
Box-Cox transformation: Lambda = 0.5
Johnson transformation function:
0.882908 + 0.987049 * Ln( ( X + 0.132606 ) / ( 9.31101 - X ) )
Goodness of Fit Test:
Distribution AD P LRT P
Normal 1.028 0.010
Box-Cox Transformation 0.301 0.574
Lognormal 1.477 <0.005
3-Parameter Lognormal 0.523 * 0.007
Exponential 5.982 <0.003
2-Parameter Exponential 3.892 <0.010 0.000
Weibull 0.248 >0.250
3-Parameter Weibull 0.359 0.467 0.225
Smallest Extreme Value 3.410 <0.010
Largest Extreme Value 0.504 0.213
Gamma 0.489 0.238
3-Parameter Gamma 0.547 * 0.763
Logistic 0.879 0.013
Loglogistic 1.239 <0.005
3-Parameter Loglogistic 0.692 * 0.085
Johnson Transformation 0.231 0.799
Best fit
distribution will
be having p–
value greater
than 0.05. But
The Best fit is
Johnson
Transformation
.

NON–NORMAL DATA75

NON–NORMAL DATA76

EXERCISE (Minitab)–BOTH SUPPLIERS77
Histogram of camshaft length suggests mixed
populations.
Further investigation revealed that there are two
suppliers for the camshaft. Data was collected
over camshafts from both sources.
Are the two suppliers similar in performance?
If not, What are your recommendations?
FILENAME: “camshafts.mtw”

MODULE # 5
ANALYSIS PHASE
 Testing of Hypothesis for Variable data
 Testing of Hypothesis for Attribute data
 Scatter Plot
 Linear Regression & Correlation

STATISTICAL METHODS
Statistical
Methods
Descriptive
Statistics
Inferential
Statistics
Estimation
Hypothesis
Testing
79
Inferential Statistics – To draw inferences about the process or
population being studied by modeling patterns of data in a way that
account for randomness and uncertainty in the observations. 2
2. Wikipedia.com

HYPOTHESIS TESTING
80
Population
I believe the
population mean
age is 50
(hypothesis).
Mean
X = 20
Reject
hypothesis! Not
close.
Random
sample







 
Hypothesis means A Belief
about a Population Parameter

NULL & ALTERNATIVE
HYPOTHESIS81
The hypotheses to be tested consists of two complementary statements:
1) The null hypothesis (denoted by H0) is a statement about the value of a
population parameter; it must contain the condition of equality.
2) The alternative hypothesis (denoted by H1) is the statement that must be true
if the null hypothesis is false. e.g.:
H0: μ = some value vs H1: μ ≠ some value
H0: μ ≤ some value vs H1: μ > some value
H0: μ ≥ some value vs H1: μ < some value
What Do You Do? If You have:
 Different types of Materials. (Stainless, Carbon Steel & Aluminum)
 Different types of oils. (Shell & Mobil)
 Different type of Cleaning solutions. (Hydrocarbon & Water base)
You want to find which method of cleaning yield the best results for all
these materials?

SAMPLING RISK
82  α - Risk, also referred as Type I Error or Producer’s Risk:
 Is the risk of rejecting H0 when H0 is true.
 i.e. concluding that the process has drifted when it really has not.
 β - Risk, also referred to Type II Error or Consumer’s Risk:
 Is the risk of accepting H0 when H0 is false.
 i.e. failing to detect the drift that has occurred in a process.
HYPOTHESIS STATEMENT:
H0 : μ = some value
H1 : μ ≠ some value
Criteria for “Accepting” & “Rejecting” a Null
Hypothesis: 1. For any fixed α, an increase in
the sample size will cause a
decrease in β.
2. For any fixed sample size, a
decrease in α will cause an
increase in β. Conversely, an
increase in α will cause a
decrease in β.
3. To decrease both α and β,
increase the sample size.

What is P-Value?83
This is the probability that a value as extreme as X–Bar (i.e.
≥ X–Bar) is observed, given that H0 is true. We reject H0 if
the obtained P-Value is less than α.
Interpreting P-Value:
H0 : μ = 5
H1 : μ ≠ 5
α = 0.05
A low p-value for the statistical test points to
rejection of Null hypothesis because it indicates
how unlikely it is that a test statistic as extreme as
or more extreme than a observed from this
population if Null Hypothesis is true.
If a p-value = 0.005, this means that if the
population means were equal (as hypothesized),
there is only 5 in 1000 chance that a more
extreme test statistic would be obtain using data
from this population and there is significant
evidence to support the Alternative Hypothesis
(H1).
P-value ≥ α, Accept Ho
P-value < α, Reject Ho

ANALYZE PHASE
HYPOTHESIS TESTING FOR
CONTINUOUS DATA
84

STATISTICAL INFERENCE
85
Comparing Two
groups
Data Normally
Distributed
Equality of Variances
Equal Variances if P
≥ 0.05
Unequal Variances if
P<0.05
Indep. Samp. T Tests Indep. Samp. T Tests
(Weltch
Approximation)
Comparing one
group with a Target
One Sample
Measured once
One Sample
Measured Twice
Data Distribution
Normal Data (P≥0.05)
One sample T Test
Data Distribution
Normal (P≥0.05)
Paired Sample T-Test
Comparing More
than Two groups
Data Distribution
One Way Anova Test *Welch Test
Testing of Hypothesis
Decision Making
1.Data is Normal when p ≥ 0.05 ,Use Anderson test
2.The Variance of groups are equal when p ≥ 0.05 Use the Levenes Test
3.Accept the Null Hypothesis when P≥0.05 otherwise accept the alternative hypothesis
Levenes Test
Normal (P≥0.05)
≥ 0.05
P<0.05
Levenes Test
* Not Available in Minitab
Testing of Hypothesis for Variable Data

TEST OF MEANS (t-tests): 1 Sample t
Measurements were made on nine widgets. You know that the distribution of
widget measurements has historically been close to normal, but suppose that
you do not know Population Standard deviation. To test if the population mean
is 5 and to obtain a 90% confidence interval for the mean, you use a t-
procedure.
1. Open the worksheet EXH_STAT.MTW.
2. Check the Normality of the data using Normality Test “VALUE”.
3. Choose Stat > Basic Statistics > 1-Sample t.
4. In Samples in columns, enter Values.
5. Check Perform hypothesis test. In Hypothesized mean, enter 5.
6. Click Options. In Confidence level, enter 90. Click OK in each dialog
box.
86
Target
A 1-sample t-test is used to compare an
expected population Mean to a target.
μsample

TEST OF MEANS (t-tests): 2-Sample
(Independent) t Test
Practical Problem:
We have conducted a study in order to determine the effectiveness of a new heating
system. We have installed two different types of dampers in home ( Damper = 1 and
Damper = 2).
We want to compare the BTU.In data from the two types of dampers to determine if
there is any difference between the two products.
 Open the MINITABTM worksheet: “Furnace.MTW”
Statistical Problem:
Ho:μ1 = μ2
Ha:μ1 ≠ μ2
2-Sample t-test (population Standard Deviations unknown).
α = 0.05
87
No, not that kind of damper!

2-Sample (Independent) t Test:
Follow the Roadmap…88
NORMALITY TEST

Follow the Roadmap…
89
TEST OF EQUAL VARIANCE
Stat ANOVA  Test for
Equal Variances…
Damper
95% Bonferroni Confidence Intervals for StDevs
2
1
4.03.53.02.52.0
Damper
BTU.In
2
1
2015105
F-Test
0.996
Test Statistic 1.19
P-Value 0.558
Levene's Test
Test Statistic 0.00
P-Value
Test for Equal Variances for BTU.In
Sample 1
Sample 2

Equal Variance90
There is no difference between
the dampers for BTU’s in.

2-Sample (Independent) t
Test: EXERCISE91
A bank with a branch located in a commercial district of a city has the business
objective of developing an improved process for serving customers during the
noon- to-1 P.M. lunch period. Management decides to first study the waiting
time in the current process. The waiting time is defined as the time that
elapses from when the customer enters the line until he or she reaches the
teller window. Data are collected from a random sample of 15 customers, and
the results (in minutes) are as follows (and stored in Bank-I):
4.21 5.55 3.02 5.13 4.77 2.34 3.54 3.20
4.50 6.10 0.38 5.12 6.46 6.19 3.79
Suppose that another branch, located in a residential area, is also concerned
with improving the process of serving customers in the noon-to-1 P.M. lunch
period. Data are collected from a random sample of 15 customers, and the
results are as follows (and stored in Bank-II):
9.66 5.90 8.02 5.79 8.73 3.82 8.01 8.35
10.49 6.68 5.64 4.08 6.17 9.91 5.47
Is there evidence of a difference in the mean waiting time between the two
branches? (Use level of significance = 0.05)

PARAMETRIC STATISTICAL
INFERENCE92
Comparing Two
groups
Data Normally
Distributed
≥ 0.05
P<0.05
Indep. Samp. T Tests Indep. Samp. T Tests
(Weltch
Approximation)
Comparing one
group with a Target
One Sample
Measured once
One Sample
Measured Twice
Data Distribution
Normal Data (P≥0.05)
One sample T Test
Data Distribution
Normal (P≥0.05)
Paired Sample T-Test
Comparing More
than Two groups
Data Distribution
One Way Anova Test *Welch Test
Testing of Hypothesis
Decision Making
1.Data is Normal when p ≥ 0.05 ,Use Anderson test
2.The Variance of groups are equal when p ≥ 0.05 Use the Levenes Test
3.Accept the Null Hypothesis when P≥0.05 otherwise accept the alternative hypothesis
Levenes Test
Normal (P≥0.05)
≥ 0.05
P<0.05
Levenes Test
* Not Available in Minitab

TEST OF MEANS (t-tests):
PAIRED T-TEST
Practical Problem:
 We are interested in changing the sole material for a popular brand of
shoes for children.
 In order to account for variation in activity of children wearing the
shoes, each child will wear one shoe of each type of sole material. The
sole material will be randomly assigned to either the left or right shoe.
Statistical Problem:
Ho: μδ = 0
Ha: μδ ≠ 0
Paired t-test (comparing data that must remain paired).
α = 0.05
93
Just checking your souls,
er…soles!
EXH_STAT.MTW

PAIRED T-TEST94
NORMALITY TEST: “Delta”
Calc Calculator
AB Delta
Percent
1.51.00.50.0-0.5
99
95
90
80
70
60
50
40
30
20
10
5
1
Mean
0.622
0.41
StDev 0.3872
N 10
A D 0.261
P-Value
Probability Plot of AB Delta
Normal

PAIRED T-TEST
Analyze this data is to use the paired t-test command.
95
Stat  Basic Statistics  Paired T-test
Paired T-Test and CI: Mat-A, Mat-B
Paired T for Mat-A - Mat-B
N Mean StDev SE Mean
Mat-A 10 10.6300 2.4513 0.7752
Mat-B 10 11.0400 2.5185 0.7964
Difference 10 -0.410000 0.387155 0.122429
95% CI for mean difference: (-0.686954, -0.133046)
T-Test of mean difference = 0 (vs not = 0): T-Value = -3.35 P-Value = 0.009
The P-value of from this
Paired T-Test tells us the
difference in materials is
statistically significant.

EXERCISE: PAIRED T-TEST
96
Nine experts rated two brands of Colombian coffee in a taste-testing experiment. A
rating on a 7- point scale (1 = extremely unpleasing, 7 = extremely pleasing) is given
for each of four characteristics: taste, aroma, richness, and acidity. The following
data (stored in coffee) display the ratings accumulated over all four characteristics.
Brand
Expert A B
C.C. 24 26
S.E. 27 27
E.G. 19 22
B.L. 24 27
C.M. 22 25
C.N. 26 27
G.N. 27 26
R.M. 25 27
P.V. 22 23
At the 0.05 level of significance, is there evidence of a difference in the mean
ratings between the two brands?

ANOVA: EXAMPLE
We have three potential suppliers that claim to have equal levels of quality. Supplier
B provides a considerably lower purchase price than either of the other two vendors.
We would like to choose the lowest cost supplier but we must ensure that we do not
effect the quality of our raw material.
97
Supplier A Supplier B Supplier C
3.16 4.24 4.58
4.35 3.87 4.00
3.46 3.87 4.24
3.74 4.12 3.87
3.61 3.74 3.46
We would like test the data to determine whether there is a difference between the
three suppliers.
TEST FOR MORE THAN TWO MEANS
(F – Test): ANOVA...

FOLLOW THE ROADMAP…TEST FOR NORMALITY
98
Supplier C
Percent
5.04.54.03.53.0
99
95
90
80
70
60
50
40
30
20
10
5
1
Mean
0.910
4.03
StDev 0.4177
N 5
AD 0.148
P-Value
Probability Plot of Supplier C
Normal
Supplier B
Percent
4.504.254.003.753.50
99
95
90
80
70
60
50
40
30
20
10
5
1
Mean
0.385
3.968
StDev 0.2051
N 5
AD 0.314
P-Value
Probability Plot of Supplier B
Normal
Supplier A
Percent
4.54.03.53.02.5
99
95
90
80
70
60
50
40
30
20
10
5
1
Mean
0.568
3.664
StDev 0.4401
N 5
AD 0.246
P-Value
Probability Plot of Supplier A
Normal All three suppliers samples are
Normally Distributed.
Supplier A P-value = 0.568
Supplier B P-value = 0.385
Supplier C P-value = 0.910

TEST FOR EQUAL VARIANCE
STACK DATA FIRST:
Data  stack  Columns…
99

TEST FOR EQUAL
VARIANCE…
100

ANOVA Using Minitab
101
Click on “Graphs…”,
Check “Boxplots of data”
Data
Supplier CSupplier BSupplier A
4.6
4.4
4.2
4.0
3.8
3.6
3.4
3.2
3.0
Boxplot of Supplier A, Supplier B, Supplier C

ANOVA: Session window
102
Test for Equal Variances: Suppliers vs ID
One-way ANOVA: Suppliers versus ID
Analysis of Variance for Supplier
Source DF SS MS F P
ID 2 0.384 0.192 1.40 0.284
Error 12 1.641 0.137
Total 14 2.025
Individual 95% CIs For Mean
Based on Pooled StDev
Level N Mean StDev ----------+---------+---------+------
Supplier 5 3.6640 0.4401 (-----------*-----------)
Supplier 5 3.9680 0.2051 (-----------*-----------)
Supplier 5 4.0300 0.4177 (-----------*-----------)
----------+---------+---------+------
Pooled StDev = 0.3698 3.60 3.90 4.20
Normal data P-value > .05
No Difference

ANOVA Assumptions
In one-way ANOVA, model adequacy can be checked by either of the following:
1. Check the data for Normality at each level and for homogeneity of variance across all
levels.
2. Examine the residuals (a residual is the difference in what the model predicts and the
true observation).
i. Normal plot of the residuals
ii. Residuals versus fits
iii. Residuals versus order
103

104
Residual
Frequency
0.60.40.20.0-0.2-0.4-0.6
5
4
3
2
1
0
Histogram of the Residuals
(responses are Supplier A, Supplier B, Supplier C)
The Histogram of
residuals should show a
bell shaped curve.
ANOVA Assumptions
Residual
Percent
1.00.50.0-0.5-1.0
99
95
90
80
70
60
50
40
30
20
10
5
1
Normal Probability Plot of the Residuals
 Normality plot of the
residuals should follow a
straight line.
 Results of our example look
good.
 The Normality assumption is
satisfied.

105
Fitted Value
Residual
4.054.003.953.903.853.803.753.703.65
0.75
0.50
0.25
0.00
-0.25
-0.50
Residuals Versus the Fitted Values
 The plot of residuals versus fits examines constant variance.
 The plot should be structureless with no outliers present.
 Our example does not indicate a problem.
ANOVA Assumptions

ANOVA EXERCISE
EXERCISE OBJECTIVE: Utilize what you have learned to
conduct and analyze a one way ANOVA using MINITABTM.
You design an experiment to assess the durability of four experimental
carpet products. You place a sample of each of the carpet products in
four homes and you measure durability after 60 days. Because you
wish to test the equality of means and to assess the differences in
means, you use the one-way ANOVA procedure (data in stacked form)
with multiple comparisons. Generally, you would choose one multiple
comparison method as appropriate for your data.
1. Open the worksheet EXH_AOV.MTW.
2. Choose Stat > ANOVA > One-Way.
3. In Response, enter Durability. In Factor, enter Carpet.
4. Click OK in each dialog box.
106

HYPOTHESIS TESTING ROADMAP
ATTRIBUTE DATA
Attribute Data
One Factor Two Factors
One Sample
Proportion
Two Sample
Proportion
MINITABTM:
Stat - Basic Stats - 2 Proportions
If P-value < 0.05 the proportions
are different
Chi Square Test
(Contingency Table)
MINITABTM:
Stat - Tables - Chi-Square Test
If P-value < 0.05 the factors are not
independent
Chi Square Test
(Contingency Table)
MINITABTM:
If P-value < 0.05 at least one
proportion is different
Two or More
Samples
Two
SamplesOne Sample
107

PROPORTION VERSUS A TARGET
This test is used to determine if the process proportion (p) equals some
desired value, p0.
The hypotheses:
 H0: p = p 0
 Ha: p ¹ p 0
The observed test statistic is calculated as follows: (normal approximation)
 
  np1p
pp
Z
00
0
obs



ˆ
ONE SAMPLE PROPORTION (Z – Test)
108

A county district attorney would like to run for the office of state district attorney. She has
decided that she will give up her county office and run for state office if more than 65% of her
party constituents support her. You need to test H0: p = .65 versus H1: p > .65.
As her campaign manager, you collected data on 950 randomly selected party members and
find that 560 party members support the candidate. A test of proportion was performed to
determine whether or not the proportion of supporters was greater than the required
proportion of 0.65. In addition, a 95% confidence bound was constructed to determine the
lower bound for the proportion of supporters.
1. Choose Stat > Basic Statistics > 1 Proportion.
2. Choose Summarized data.
3. In Number of events, enter 560. In Number of trials, enter 950.
4. Check Perform hypothesis test. In Hypothesized proportion, enter 0.65.
5. Click Options. Under Alternative, choose greater than. Click OK in each dialog
box.
As P-Value > 0.05, mean accept H0; means
As her campaign manager, you would
advise her not to run for the office of state
district attorney.
109

EXAMPLE: You are the shipping manager and are in charge of improving
shipping accuracy. Your annual bonus depends on your ability to prove
that shipping accuracy is better than the target of 80%.
Out of 2000 shipments only 1680 were accurate.
• Do you get your annual bonus?
 Choose Stat  Basic Statistics  1 Proportion
 Choose Summarized data.
110

ATTRIBUTE DATA
Attribute Data
One Sample
Proportion
Two Sample
Proportion
MINITABTM:
are different
Chi Square Test
(Contingency Table)
MINITABTM:
independent
Chi Square Test
(Contingency Table)
MINITABTM:
Two or More
Samples
Two
SamplesOne Sample
111

COMPARING TWO PROPORTIONS
This test is used to determine if the process defect rate (or
proportion, p) of one sample differs by a certain amount ‘D’
from that of another sample (e.g., before and after your
improvement actions)
The hypotheses:
H0: p1 - p2 = D
Ha: p1 - p2 = D
The test statistic is calculated as follows:
    222111
21
obs
npˆ1pˆnpˆ1pˆ
Dpˆpˆ
Z



TWO SAMPLE PROPORTIONS (Z – Test)
112

Hypotheses:
 H0: p1 – p2 = 0.0
 Ha: p1 – p2 = 0.0
Two sample proportion test
 Choose level of Significance = 5%
COMPARING TWO PROPORTIONS: EXAMPLE
As your corporation's purchasing manager, you need to authorize the purchase of
twenty new photocopy machines. After comparing many brands in terms of price,
copy quality, warranty, and features, you have narrowed the choice to two: Brand
X and Brand Y. You decide that the determining factor will be the reliability of the
brands as defined by the proportion requiring service within one year of purchase.
Because your corporation already uses both of these brands, you were able to
obtain information on the service history of 50 randomly selected machines of
each brand. Records indicate that six Brand X machines and eight Brand Y
machines needed service. Use this information to guide your choice of brand for
purchase.
113

COMPARING TWO PROPORTIONS
 Choose Stat > Basic Statistics > 2 Proportions.
 Choose Summarized data.
 In First sample, under Events, enter 44. Under Trials, enter 50.
 In Second sample, under Events, enter 42. Under Trials, enter 50. Click
OK.
As Both P – Value > 0.05, Accepts H0; means the proportion of photocopy
machines that needed service in the first year did not differ depending on
brand. As the purchasing manager, you need to find a different criterion to
guide your decision on which brand to purchase.
114

ATTRIBUTE DATA
Attribute Data
One Sample
Proportion
Two Sample
Proportion
MINITABTM:
are different
Chi Square Test
(Contingency Table)
MINITABTM:
independent
Chi Square Test
(Contingency Table)
MINITABTM:
Two or More
Samples
Two
SamplesOne Sample
115

TWO OR MORE SAMPLE PROPORTIONS
(Chi–Square Test) “ONE FACTOR”….
Contingency Tables
The null hypothesis is that the population proportions of each group are the same.
 H0: p1 = p2 = p3 = … = pn
 Ha: at least one p is different
Statisticians have shown that the following statistic forms a chi-square distribution when
H0 is true:
Where “observed” is the sample frequency, “expected” is the calculated frequency based
on the null hypothesis, and the summation is over all cells in the table.
 

expected
expectedobserved
2
 


c
1j ij
2
ijij
r
1i
2
o
E
)E(O
χ
Chi-square Test:
Test Statistic Calculations
116

TWO OR MORE SAMPLE PROPORTIONS
(Chi–Square Test) “ONE FACTOR”….
Contingency Tables
117
The political affiliation of a certain city's population is: Republicans 52%, Democrats
40%, and Independent 8%. A local university student wants to assess if the
political affiliation of the university students is similar to that of the population. The
student randomly selects 200 students and records their political affiliation.
1. Open the worksheet POLL.MTW.
2. Choose Stat  Tables  Chi-Square Goodness-of-Fit Test (One Variable).
As P-Value < 0.05; Reject H0; means the
political affiliation of the university
students is not the same as those of the
population.

ATTRIBUTE DATA
Attribute Data
One Sample
Proportion
Two Sample
Proportion
MINITABTM:
are different
Chi Square Test
(Contingency Table)
MINITABTM:
independent
Chi Square Test
(Contingency Table)
MINITABTM:
Two or More
Samples
Two
SamplesOne Sample
118

Test for association (or dependency) between two
classifications
(Chi–Square Test) “TWO FACTORS”….
Contingency Tables
119
Exercise objective: To practice solving problem presented using
the appropriate Hypothesis Test.
 You are the quotations manager and your team thinks that the reason
you don’t get a contract depends on its complexity.
 You determine a way to measure complexity and classify lost contracts
as follows:
1. Write the null and alternative hypothesis.
2. Does complexity have an effect?
Low Med High
Price 8 10 12
Lead Time 10 11 9
Technology 5 9 16

Test for association (or dependency) between
two classifications
Contingency Tables
120
First we need to create a table in
MINITABTM
Secondly, in MINITABTM perform a
Chi-Square Test

Test for association (or dependency) between
two classifications
Contingency Tables
121
Are the factors independent of
each other?
Yes; Both factors are independent

ANALYZE PHASE…
SCATTER PLOT, CORRELATION
AND
SIMPLE & MULTIPLE
REGRESSION ANALYSIS
122

SCATTER PLOT
WHAT IS A SCATTER PLOT?
Is a graphical presentation of any possible relationship between
two sets of variables by a simple X-Y plot, which may or may
not be dependent.
123
EXAMPLE: You are interested in how well your company's camera batteries are
meeting customers' needs. Market research shows that customers become annoyed
if they have to wait longer than 5.25 seconds between flashes.
You collect a sample of batteries that have been in use for varying amounts of time
and measure the voltage remaining in each battery immediately after a flash
(VoltsAfter), as well as the length of time required for the battery to be able to flash
again (flash recovery time, FlashRecov). Create a scatter plot to examine the
results. Include a reference line at the critical flash recovery time of 5.25 seconds.
Open the worksheet BATTERIES.MTW

SCATTER PLOT
INTERPRETING THE RESULTS:
As expected, the lower the voltage in a battery after a
flash, the longer the flash recovery time tends to be.
The reference line helps to illustrate that there were many
flash recovery times greater than 5.25 seconds.
125

CORRELATION
 Correlation analysis is a method that is used to measure the strength of
the linear relationship between two or more continuous variables.
126
 When ‘r’ is close to +1, there is a strong positive correlation.
 When ‘r’ is close to Zero, there is very little or no correlation.
 When ‘r’ is close to –1, there is a strong negative correlation.
H0: There is No Correlation
H1: There is Correlation

SCATTER PLOT & CORRELATION
EXAMPLE: You are interested in how well your company's
camera batteries are meeting customers' needs. Market
research shows that customers become annoyed if they have to
wait longer than 5.25 seconds between flashes.
You collect a sample of batteries that have been in use for
varying amounts of time and measure the voltage remaining in
each battery immediately after a flash (VoltsAfter), as well as
the length of time required for the battery to be able to flash
again (flash recovery time, FlashRecov). Create a scatter plot to
examine the results. Include a reference line at the critical flash
recovery time of 5.25 seconds.
Open the worksheet BATTERIES.MTW
127

SCATTER PLOT & CORRELATION
EXAMPLE…:
Correlations: FlashRecov, VoltsAfter
Pearson correlation of FlashRecov and VoltsAfter = -0.478
P-Value = 0.002
GENERAL GUIDELINE:
+/-0.8 imply a good
correlation
Hypothesized statement:
H0 : No correlation between the 2 variables
H1 : Significant correlation between the 2 variable
128

SCATTER PLOT &
CORRELATION…
EXERCISE#2:
The following information taken
from annual report of a company
shows net sales (NS) & working
capital (WC).
a) Plot the variable NS and WC in
scatter plot. Format, what, if
any kind of relationship
appears to exist between
them.
b) Compute the correlation
coefficient between NS & WC.
YEAR NET SALES WORKING
CAPITAL
1989 234463 67168
1990 281462 69788
1991 294030 75306
1992 286495 84740
1993 318930 97343
1994 356595 108601
1995 418152 118550
1996 473103 145069
1997 502875 146975
1998 557840 141268
129

WHAT IS REGRESSION?
Method of determining the statistical relationship between a response
(or output) and one or more predictor (or input) variables.
Y = ƒ (X1, X2, . . . . Xn)
Where ‘Y’ is the RESPONSE and X1 to Xn are the PREDICTORS
130
Simple Linear Regression…
Is when the dependent variable is linearly proportional to just ONE
independent variable.
Multiple Regression…
May be viewed as an extension of simple regression analysis (where only
one predictor is involved) to the situation where there is more than ONE
predictor to be considered.
TYPES OF REGRESSION

SIMPLE LINEAR
REGRESSION
EXAMPLE:
A study was conducted with
vegetarians to see whether the
number of grams of protein
each ate per day was related to
diastolic blood pressure. The
data are given here. If there is
a significant relationship,
predict the diastolic pressure of
a vegetarian who consumers 8
grams of protein per day.
131

SIMPLE LINEAR
REGRESSIONEXAMPLE…
Regression Analysis: Pressure versus Grams
The regression equation is
Pressure = 64.9 + 2.66 Grams
Predictor Coef SE Coef T P
Constant 64.936 3.401 19.09 0.000
Grams 2.6623 0.4408 6.04 0.001
S = 2.84522 R-Sq = 83.9% R-Sq(adj) = 81.6%
Analysis of Variance
Source DF SS MS F P
Regression 1 295.33 295.33 36.48 0.001
Residual Error 7 56.67 8.10
Total 8 352.00
REGRESSION EQUATION
Coefficient of Determination-”R-Sq” is
the measure of the fit of the Regression
to the data. It suggest a very good fit
when R-Sq approach 100%
The F-test is a test of the hypothesis……
H0: All Regression coefficients, except b0 are Zero
H1: The Regression is Statistically significant
132

SIMPLE LINEAR
REGRESSION
EXERCISE:
A study is conducted to
determine the relationship
between a driver’s age and
the number of accidents he
or she has over a one-year
period. If there is a
significant relationship,
predict the number of
accidents of a driver who is
28.
Driver’s
Age
No. of
accidents
16 3
24 2
18 5
17 2
23 0
27 1
32 1
133

MULTIPLE LINEAR
REGRESSION134
(MLR) model: Y = β0 + β1 X1 + β2 X2 …….
Where X’S is the predictor (independent) variables
Y is the response (dependent) variable
β0 is the intercept
β1, β2… are the slopes for the respective predictors
EXAMPLE: As part of a test of solar thermal energy, we need to measure
the total heat flux from homes. We wish to examine whether total heat flux
(HeatFlux) can be predicted by the position of the focal points in the east,
south, and north directions.
We will evaluate the three-predictor (three input variables; east, south and
north) model using multiple regression.
1. Open the worksheet EXH_REGR.MTW
2. Choose Stat > Regression > Regression.
3. In Response, enter HeatFlux.
4. In Predictors, enter East South North.

MULTIPLE LINEAR REGRESSION
135 EXAMPLE…:
Results for: Exh_regr.MTW
Regression Analysis: HeatFlux versus East, South, North
The regression equation is
HeatFlux = 389 + 2.12 East + 5.32 South - 24.1 North
Predictor Coef SE Coef T P
Constant 389.17 66.09 5.89 0.000
East 2.125 1.214 1.75 0.092
South 5.3185 0.9629 5.52 0.000
North -24.132 1.869 -12.92 0.000
S = 8.59782 R-Sq = 87.4% R-Sq(adj) = 85.9%
The p-values for the estimated
coefficients of North and South are
both 0.000, indicating that they are
significantly related to HeatFlux.
The R-Square value indicates
87.4% of the variance in HeatFlux
is due to the predictors

MODULE # 6
IMPROVE PHASE
 Design of Experiment: An Introduction
 2K Factorial Design

PROJECT STATUS REVIEW
137
1. Understand our problem and it’s impact on the business.
(DEFINE)
2. Established firm objectives / goals for improvement.
(DEFINE)
3. Quantified our output characteristic. (DEFINE)
4. Validated the measurement system for our output
characteristic. (MEASURE)
5. Identified the process input variables in our process.
(Measure)
6. Narrowed our input variables to the potential “X’s” through
statistical Analysis. (ANALYZE)
7. Selected the Vital few X’s to optimize the output response(s).
(IMPROVE)
8. Quantified the relationship of the Y’s to the X’s with Y = f(x).
(IMPROVE)

WHAT IS EXPERIMENT?
138
 In statistics, an experiment refers to any process that generates a set of
data.
 An experiment involves a test or series of test in which purposeful
changes are made to the input variables of a process or system so that
changes in the output responses can be observed and identified.
Noise Factors

TERMINOLOGIES
139
Terms used in Design of Experiments (DOE) need to defined, these are:
 RESPONSE:
 A measurable outcome of interest, e.g.: yield, strength, etc.
 FACTORS:
 Controllable variables that are deliberately manipulated to determine
their individual and joint effects on the response(s), OR Factors are
those quantities that affect the outcome of an experiment, e.g.:
temperature, time, etc.
 LEVELS:
 Levels refer to the values of factors for which the data is gathered,
“values that factor will take in an experiment”, e.g.:
Level–1 for time = 2hours
Level–2 for time = 3 hours
 TREATEMENT:
 A set of specified factor levels for an experimental run, e.g.:
Treatment–1: time = 2hrs and temperature = 1750 C
Treatment–2: time = 3hrs and temperature = 2250 C

EXAMPLES
140
 EXAMPLE–1:
 In a MACHING PROCESS
 RESPONSE: Surface Finish “Y”
 FACTORS: Speed of machine “XA” & Depth of Cut “XB”
 LEVELS: High & Low
 EXAMPLE–2:
 In a POPCORN MAKING PROCESS
 RESPONSE: Volume (ml) Yield of Popcorn “Y”
 FACTORS: Type of Popper “XA” & Grade of corn used
“XB”
 LEVELS: Air, and Oil & Budget, Regular and luxury

TYPES OF EXPERIMENTS
141
EXPERIMENTS
ONE-FACTOR AT A TIME
EXPRIMENTS
BEST GUESS
EXPERIMENTS
FACTORIAL
EXPERIMENTS

2K FACTORIAL
142
 2K Factorial Designs are experiments where all
FACTORS have only TWO LEVELS
 The number of combinations (Runs) for Full
Factorial Design is denoted as n = 2k (where
k=number of Factors)
2K
Factors
Levels

22 FACTORIAL
EXPERIMENTAL DESIGN143
EXAMPLE: Consider the manufacture of a product, for use
in the making of paint, in a batch process. Fixed amounts of raw
material are heated under pressure in rector-1 for a fixed period
of time and the product is then recovered. Currently the process
is operated at temperature 225o C and pressure 4.5 bar. As part
of Six Sigma project, aimed at increasing product yield, a 22
factorial experiment with two replications was planned. Yields
are typically around 90 Kg. It was decided after discussion
amongst the project team to use the levels 200o C and 250o C
for temperature and level 4.0 bar and 5.0 bar for pressure.
 RESPONSE: Product Yield “Y”
 FACTORS: Temperature “XA” & Pressure “XB”
 LEVELS: 200o C and 250o C & 4.0 bar and 5.0
bar

22 FACTORIAL
EXAMPLE…:

22 FACTORIAL
EXAMPLE…:
Stat > DOE > Factorial > Factorial Plots

22 FACTORIAL
EXAMPLE…:
The Main Effect Plot indicate that:
 On average, increasing temperature
from 200o C to 250o C increases yield
of product by 8 kg.
 On average, increasing pressure from 4
bar to 5 bar decreases yield of product
by 6Kg.
The parallel lines indicate no temperature–
Pressure interaction here.

22 FACTORIAL
EXAMPLE…:
Stat > DOE > Factorial > Analyze Factorial Design…
Factorial Fit: Yield versus Temperature, Pressure
Estimated Effects and Coefficients for Yield (coded units)
Term Effect Coef SE Coef T P
Constant 92.000 0.9354 98.35 0.000
Temperature 8.000 4.000 0.9354 4.28 0.013
Pressure -6.000 -3.000 0.9354 -3.21 0.033
Temperature*Pressure 0.000 -0.000 0.9354 -0.00 1.000
S = 2.64575 PRESS = 112
R-Sq = 87.72% R-Sq(pred) = 50.88% R-Sq(adj) = 78.51%
The P–Value indicate
that both temperature &
pressure have a real
effect on Yield.

22 FACTORIAL
EXAMPLE…:

22 FACTORIAL
EXERCISE:
An Engineer desire to study which is
the 2 Factors determined that affect
the Defect Rate in his production
line.
FACTORS:
Temperature & Pressure
LEVELS:
Temperature – 60 & 70o C &
Pressure – 3.0 & 5.5 Bar
REPLICATES: 3
DEFECT
3.93183
2.30259
0.0000
2.07944
4.33073
3.33220
2.39790
0.69315
2.19722
2.83321
1.38629
1.38629

23 FACTORIAL
EXAMPLE: A plastic manufacturing company had formed a
work improvement company had formed a work
improvement team consisting of engineers from different
department. The team objective is to strive to improve the
yield of a coating process. After a series of brainstorming
session, the team determined that the following are the
deciding factors and levels:
A: Temperature: 400o F and 450o F
B: Catalyst Con.: 10% and 20%
C: Processing Ramp time: 45 seconds and 90
seconds
The design is a 23 factorial and each run (treatment) is
replicated 3 times and total is 24 randomized trial.

23 FACTORIAL
EXAMPLE (Cont…):

23 FACTORIAL
EXAMPLE (Cont…):
Stat > DOE > Factorial > Factorial Plots

23 FACTORIAL
EXAMPLE (Cont…):

23 FACTORIAL
EXAMPLE (Cont…):
Stat > DOE > Factorial > Analyze Factorial Design…

23 FACTORIAL
EXAMPLE (Cont…):

MODULE # 7
CONTROL PHASE
 SPC: An Introduction
 Attribute Control Charts
 Variable Control Charts
 Control Plan

INTRODUCTION TO SPC
 In 1924, Shewhart applied the terms of "assignable-cause" and "chance-cause"
variation and introduced the "control chart" as a tool for distinguishing between
the two.
 Central to an SPC program are the following:
 Understand the causes of variability:
 Shewhart found two basic causes of variability:
 Chance causes of variability
 Assignable causes of variability
158
 OBJECTIVES OF SPC CHARTS
 All control charts have one primary purpose!
 To detect assignable causes of variation that cause significant process shift, so that:
 investigation and corrective action may be undertaken to rid the process of the
assignable causes of variation before too many nonconforming units are
produced. In other words, to keep the process in statistical control.
 The following are secondary objectives or direct benefits of the primary objective:
 To reduce variability in a process.
 To Help the process perform consistently & predictably.
 To help estimate the parameters of a process and establish its process capability.

CONTROL CHART ANATOMY
159
Common Cause
Variation
Process is “In
Control”
Special Cause
Variation
Process is “Out
of Control”
Special Cause
Variation
Process is “Out
of Control”
Run Chart of
data points
Process Sequence/Time Scale
Lower Control
Limit
Mean
+/-3sigma
Upper Control
Limit

INTERPRETING CONTROL CHART
160

TYPES & SELECTION OF
CONTROL CHART161
What type of
data do I
have?
Variable Attribute
Counting defects
or defectives?
X-bar &
S Chart
I & MR
Chart
X-bar &
R Chart
n > 10 1 < n < 10 n = 1
Defectives Defects
What subgroup
size is available?
Constant
Sample Size?
Constant
Opportunity?
yes yesno no
P or np
Chart
u Chartp Chart c or u
Chart
Note: A defective unit can
have more than one defect.

Calculate the parameters of the “P”
Control Charts with the following:162
Where:
p: Average proportion defective (0.0 – 1.0)
ni: Number inspected in each subgroup
LCLp: Lower Control Limit on P Chart
UCLp: Upper Control Limit on P Chart
inspecteditemsofnumberTotal
itemsdefectiveofnumberTotal
p 
in
pp )1(
3pUCLp


Center Line Control Limits
in
pp )1(
3pLCLp


Since the Control Limits are a function of sample
size, they will vary for each sample.

CONTROL CHARTS FOR ATTRIBUTE DATA
163
P Chart With constant sample size: EXAMPLE
Frozen orange juice concentrate is packed in 6- oz cardboard cans. A metal
bottom panel is attached to the cardboard body. The cans are inspected for
possible leak. 20 samplings of 50 cans/sampling were obtained. Verify if the
process is in control.
Choose Stat > Control Charts >Attributes
Charts > P

CONTROL CHARTS FOR ATTRIBUTE DATA…
164
P Chart With Variable sample size: EXAMPLE
Suppose you work in a plant that manufactures picture
tubes for televisions. For each lot, you pull some of the
tubes and do a visual inspection. If a tube has scratches on
the inside, you reject it. If a lot has too many rejects, you
do a 100% inspection on that lot. A P chart can define
when you need to inspect the whole lot.
1. Open the worksheet EXH_QC.MTW.
2. Choose Stat > Control Charts >Attributes Charts >
P.
3. In Variables, enter Rejects.
4. In Subgroup sizes, enter Sampled. Click OK.

Calculate the parameters of the “np”
Control Charts with the following:
165
Since the Control Limits AND Center Line are a function
of sample size, they will vary for each sample.
subgroupsofnumberTotal
itemsdefectiveofnumberTotal
pn  )1(3pnUCL inp ppni 
p)-p(1n3pnLCL iinp 
Where:
np: Average number defective items per subgroup
LCLnp: Lower Control Limit on nP chart
UCLnp: Upper Control Limit on nP chart

ATTRIBUTE CONTROL CHARTS …
166
NP Chart: EXAMPLE
You work in a toy manufacturing company and your job is to
inspect the number of defective bicycle tires. You inspect
200 samples in each lot and then decide to create an NP
chart to monitor the number of defectives. To make the NP
chart easier to present at the next staff meeting, you decide
to split the chart by every 10 inspection lots.
1. Open the worksheet TOYS.MTW.
2. Choose Stat > Control Charts > Attributes Charts > NP.
3. In Variables, enter Rejects.
4. In Subgroup sizes, enter Inspected.
5. Click NP Chart Options, then click the Display tab.
6. Under Split chart into a series of segments for display
purposes, choose Number of subgroups in each segment and
enter10.

Calculate the parameters of the “c”
167
subgroupsofnumberTotal
defectsofnumberTotal
c  c3cUCLc 
c3cLCLc 
Where:
c: Total number of defects divided by the total number of subgroups.
LCLc: Lower Control Limit on C Chart.
UCLc: Upper Control Limit on C Chart.

ATTRIBUTE CONTROL CHARTS …
168
C Chart: EXAMPLE
Suppose you work for a linen manufacturer. Each 100 square yards of
fabric can contain a certain number of blemishes before it is rejected. For
quality purposes, you want to track the number of blemishes per 100
square yards over a period of several days, to see if your process is
behaving predictably.
1. Open the worksheet EXH_QC.MTW.
2. Choose Stat > Control Charts > Attributes Charts > C.
3. In Variables, enter Blemish.

Calculate the parameters of the “u” Control
Charts with the following:
169
InspectedUnitsofnumberTotal
IdentifieddefectsofnumberTotal
u 
in
u
3uUCLu 
in
u
3uLCLu 
Where:
u: Total number of defects divided by the total number of units inspected.
LCLu: Lower Control Limit on U Chart.
UCLu: Upper Control Limit on U Chart.
Since the Control Limits are a function of
sample size, they will vary for each sample.

ATTRIBUTE CONTROL CHARTS
(Cont…)
170 U Chart: EXAMPLE
As production manager of a toy manufacturing company, you
want to monitor the number of defects per unit of motorized
toy cars. You inspect 20 units of toys and create a U chart to
examine the number of defects in each unit of toys. You
want the U chart to feature straight control limits, so you fix
a subgroup size of 102 (the average number of toys per
unit).
1. Open the worksheet TOYS.MTW.
2. Choose Stat > Control Charts > Attributes Charts > U.
3. In Variables, enter Defects.
4. In Subgroup sizes, enter Sample.
5. Click U Chart Options, then click the S Limits tab.
6. Under When subgroup sizes are unequal, calculate control
limits, choose Assuming all subgroups have size then enter
102.

Calculate the parameters of the X–Bar and R
171
k
x
X
k
1i
i

k
R
R
k
i
i

RAXUCL 2x 
RAXLCL 2x 
RDUCL 4R 
RDLCL 3R 
Where:
Xi: Average of the subgroup averages, it becomes the Center Line of the Control Chart
Xi: Average of each subgroup
k: Number of subgroups
Ri : Range of each subgroup (Maximum observation – Minimum observation)
Rbar: The average range of the subgroups, the Center Line on the Range Chart
UCLX: Upper Control Limit on Average Chart
LCLX: Lower Control Limit on Average Chart
UCLR: Upper Control Limit on Range Chart
LCLR : Lower Control Limit Range Chart
A2, D3, D4: Constants that vary according to the subgroup sample size
Rbar (computed above)
d2 (table of constants for subgroup size n) (st. dev. Estimate) =

Calculate the parameters of the X–Bar and S
172
k
x
X
k
1i
i

k
s
S
k
1i
i

SAXUCL 3x 
SAXLCL 3x 
SBUCL 4S 
SBLCL 3S 
Where:
Xi: Average of the subgroup averages, it becomes the Center Line of the Control Chart
Xi: Average of each subgroup
k: Number of subgroups
si : Standard Deviation of each subgroup
Sbar: The average S. D. of the subgroups, the Center Line on the S chart
UCLX: Upper Control Limit on Average Chart
LCLX: Lower Control Limit on Average Chart
UCLS: Upper Control Limit on S Chart
LCLS : Lower Control Limit S Chart
A3, B3, B4: Constants that vary according to the subgroup sample size
Sbar (computed above)
c4 (table of constants for subgroup size n) (st. dev. Estimate) =

VARIABLE CONTROL CHARTS
(Cont…)173
X–Bar & S Charts: EXAMPLE
You are conducting a study on the blood glucose levels of 9 patients
who are on strict diets and exercise routines. To monitor the mean and
standard deviation of the blood glucose levels of your patients, create
an X-Bar and S chart. You take a blood glucose reading every day for
each patient for 20 days.
1. Open the worksheet
BLOODSUGAR.MTW.
2. Choose Stat > Control Charts
> Variables Charts for
Subgroups > Xbar-S.
3. Choose All observations for a
chart are in one column,
then enter Glucoselevel.
4. In Subgroup sizes, enter 9.
Click OK.

Calculate the parameters of the Individual and
MR Control Charts with the following:
174
k
x
X
k
1i
i

k
R
RM
k
i
i

RMEXUCL 2x 
RMEXLCL 2x 
RMDUCL 4MR 
RMDLCL 3MR 
Where:
Xbar: Average of the individuals, becomes the Center Line on the Individuals Chart
Xi: Individual data points
k: Number of individual data points
Ri : Moving range between individuals, generally calculated using the difference
between each successive pair of readings
MRbar: The average moving range, the Center Line on the Range Chart
UCLX: Upper Control Limit on Individuals Chart
LCLX: Lower Control Limit on Individuals Chart
UCLMR: Upper Control Limit on moving range
LCLMR : Lower Control Limit on moving range
E2, D3, D4: Constants that vary according to the sample size used in obtaining the moving
range
MRbar (computed above)
d2 (table of constants for subgroup size n) (st. dev. Estimate) =

VARIABLE CONTROL CHARTS
(Cont…)175 I & MR Charts: EXAMPLE
As the distribution manager at a limestone quarry, you
want to monitor the weight (in pounds) and variation in the
45 batches of limestone that are shipped weekly to an
important client. Each batch should weight approximately
930 pounds. you want to examine the same data using an
individuals and moving range chart.
1. Open the worksheet EXH_QC.MTW
2. Choose Stat > Control Charts > Variables Charts for
Individuals > I-MR.
3. In Variables, enter Weight.
4. Click I-MR Options, then click the Tests tab.
5. Choose Perform all tests for special causes, then click OK in
each dialog box.

176
WHAT IS CONTROL
PLAN?
 The Control Plan describes the actions
that are required at each phase of the
process to ensure that all process
outputs will be in state of control.
 Control plan is a living document,
reflecting the current methods of
control, and measurement systems
used.
 Accessible at work station

17
7
CONTROL PLAN: EXAMPLE
What to Check? How important it is …?
How to Check?
How many & When to Check?
What to do when some thing is wrong?

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